Solve quick probability questions
Company: Optiver
Role: Software Engineer
Category: Statistics & Math
Difficulty: medium
Interview Round: Take-home Project
Answer the following rapid-fire probability questions. Show formulas or reasoning briefly:
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1) When rolling two fair six-sided dice, what is the probability distribution of the sum?
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2) From a well-shuffled standard 52-card deck, after discarding the top 10 cards unseen, what is the probability that the next card is red?
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3) When rolling two fair dice, what is the probability that the sum is 11 or 12?
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4) In a fair-coin gambler’s-ruin process, starting with bankroll 10 and goal 20 (absorbing barriers at 0 and 20, betting 1 each round), what is the probability of reaching 20 before 0?
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5) If a fair six-sided die is rolled N times, what is the expected number of occurrences of each face and the variance of that count?
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6) If four fair six-sided dice are rolled, what is the probability that the product of the outcomes is odd?
Quick Answer: This set evaluates core probability and combinatorics skills including discrete distributions, conditional probability, independence, expectation and variance, and basic stochastic processes such as gambler's ruin, situated in the Statistics & Math domain.